Significant challenges are associated with the recovery of hydrocarbon-containing substances such as crude oil from subterranean reservoirs. Subterranean reservoirs typically possess convoluted, fractured and crevassed bottom surface topographies wherein significant quantities of crude oil remain in pools that are inaccessible by conventional oil well extraction systems. Numerous strategies and technologies have been developed to increase the efficiency and extent of crude oil recovery from subterranean reservoirs. Such strategies include injecting water or steam or inert gas through well casings into the reservoirs to break up obstacles (i.e., bottom surface formations) impeding the flow of crude oil to the well, or alternatively, to reduce the viscosity of the oil to increase its flowability. Other strategies to increase the flowability of crude oil within subterranean reservoirs include applications of vibrational energies generated by: (a) seismic shock as a resulted of repeatedly dropping and raising a weight within a well casing, or (b) by lowering an ultrasonic wave generating device e.g., a transducer into a well casing and then manipulating the amplitude and frequency of the waves generated. However, significant volumes of crude oil remain inaccessible.
Dynamics of porous media is of intense research concerns in petroleum engineering, geophysics, geotechnical engineering, and civil engineering, and has been extensively studied for decades. Demands from soil mechanics, oil production, modern earthquake and offshore engineering have further motivated the research on the dynamics of fluid-saturated porous media. By introducing the assumptions that the solid skeleton of the porous medium obeys the laws of homogeneous linear elasticity and the fluid obeys Darcy's laws, Biot (1956a, J. Acoust. Soc. Am. 28:168-178; 1956b, J. Acoust. Soc. Am. 28: 179-191) formulated the governing equations for wave propagation in a fully saturated medium. Biot (1956a; 1956b) also proved the existence of two compressional waves, namely the first and second compressional waves, and one rotational wave in a porous medium fully saturated by fluid. The first compressional wave is also known as the fast wave that is very similar to the compressional wave in an elastic medium, for which the displacements of solid and fluid are in phase. The second compressional wave is usually named as slow wave that has a strongly dispersive characteristic, for which the displacements of fluid and solid are out of phase. Following Biot's theory, Vardoulakis and Beskos (1986, Mech. Comp. Mat. 5: 87-108) developed a theory describing wave propagation in a three-phase porous medium which is applicable to partially-saturated materials. White (1975, Geophysics, 40: 224-232) demonstrated that wave velocity and attenuation are substantially affected by the presence of partial saturation, depending mainly on the size of the gas pockets (saturation), frequency, permeability and porosity of the media. Bardet and Sayed (1993, Soil Dynamics and Earthquake Engineering, 12: 391-402) provided exact and approximate expressions for the velocity and attenuation of the compressional waves within nearly fully saturated poroelastic media.
Recently, numerous research works are performed to improve Blot's theory and to broaden the applications of Biot's theory. Gurevich et al. (1999, Transport in Porous Media, 36: 149-160) utilized experiment and simulation methods to verify Biot's theory. Investigation on the scattering of a fast compression wave by an inhomogeneity in a fluid-saturated medium was presented by Berryman (1985, J. Math. Physics, 26: 1408-1419), who proved that there would be three scattering waves, namely a fast compression wave, a slow compression wave, and a shear wave. The properties of elastic waves in a non-Newtonian (Maxwell) fluid-saturated porous medium were studied by Tsiklauri and Beresnev (2003, Transport in Porous Media, 53: 39-50). It is generally accepted that the wave will attenuate due to the presence of the pore fluid in the porous media. Wave velocities and attenuation are two key aspects of the waves in porous media, since they are important in analyzing the dynamic response of the media with respect to the properties of the media and the wave sources, such as viscosity, frequency and porosity. Hamidzadeh and Luo (2000, Vibration and Control of Continuous Systems 107: 39-44) investigated the dynamic response of the surface of an elastic soil medium which was excited by a vertical harmonic concentrated force by using a semi-analytical method. Based on Biot-type three-phase theory, Pham et al. (2002, Geophys. Pros. 50: 615-627) presented the wave velocities and quality factors of clay-bearing sandstones as a function of pore pressure, frequency and partial saturation. A dispersion coefficient was introduced to reflect the friction between the fluid and solid in a porous medium. Extensional wave attenuation and velocity measurements on high permeability Monterey sand were performed by the authors over a range of gas saturations for imbibition and degassing conditions. The result showed that partially-saturated sands under moderate confining pressure can produce strong intrinsic attenuation for extensional waves. It was found in the study that the velocities show a gradual decrease with increasing water saturation, followed by a sharp increase at near full saturation.
In the current literature, however, there are very few studies focusing on investigating the relative displacements between the fluid and solid in a porous medium fully-saturated by Newtonian fluid. Furthermore, the prior art in this field postulates a single energy source existing in the field being considered.
Governing Equation Development
The following nomenclature is used in the prior art section and invention disclosure sections herein:    C1, C2—refer to the amplitudes of the waves propagating in solid and fluid respectively;    dj—refers to the distance from a source to the origin;    e—refers to the volume strains of solid;    exp(•)—refers to an exponential function;    H—refers to an introduced physical parameter;    H0(1)(•)—refers to a zero-order Hankel function of the first kind;    Kb—refers to the bulk modulus of the skeletal frame;    Kf—refers to the bulk modulus of the fluid;    Ks—refers to the bulk modulus of the solid;    l—refers to a wave number;    p—refers to fluid pressure;    r—refers to the distance from a point in the field to a source;    r—refers to a radius coordinate in a polar system;    rj—refers to the distance from a point P to the jth wave sources;    sij—refers to the stresses acting on the fluid of a porous medium;    t—refers to time;    u—refers to the displacement vector of a fluid;    u0j, U0j—refer to the displacements of the and fluid of the jth source respectively (j=1, 2, . . . , n);    u0j, U0j—refer to the displacement vectors of solid and fluid excited by the jth source respectively (j=1, 2, . . . , n);    U—refers to the displacement vector of a solid;    V1—refers to the dilatation wave velocity with respect to a first compressible wave;    V2—refers to the dilatation wave velocity with respect to a second compressible wave;    Vc—refers to the ratio of H and ρ;    V—refers to the reference wave velocity;    x, y—refers to the coordinates of a Cartesian coordinate system;    zj—refers to an introduced complex variable;    α—refers to the coefficient related to porosity;    δij—is the Kronecker symbol;    ε—refers to the volume strains of a fluid    θ—refers to the angular coordinate in a polar system    μs—refers to the shear modulus of a material;    νs—refers to the Poisson ratio of a solid;    ξ—refers to the ratio between reference velocity and wave velocity;    ξI, ξII—refers to roots;    ρ—refers to a density parameter;    ρ11, ρ12, ρ22—refers to the density terms of a porous medium;    ρf—refers to the mass density of a fluid;    ρs—refers to the mass density of a solid;    σij—refers to the total stresses of a porous medium;    σijs—refers to the stresses acting on the solid frame of a porous medium;    φ—refers to the porosity of a medium;    φs,—refers to the scalar potential of a solid;    φf—refers to the scalar potential of a fluid;    ψf—refers to the vector potential of a solid;    ψs—refers to the vector potential of a fluid;    ω—refers to the frequency of a wave; and    ∇, ∇2—refers to Laplacians.
Biot's theory provides a framework for analyzing the wave propagation in porous media. In Biot's most representative papers in this field (Biot, 1956a, b), the fluid in porous medium is assumed to be compressible and may flow relative to the solid. To derive the wave equations in low frequency range, the following assumptions are made:                (1) the relative motion of the fluid in pores is a laminar flow which follows Darcy's law;        (2) the elastic wavelength of the wave traveling in the porous media is much larger than that of the unit solid-fluid element;        (3) the size of the unit element is geometrically large in comparison with that of the pores.Some other basic assumptions in elastic mechanics are also employed, such as homogeneity and isotropy of the porous media material and the impervious of the pore wall, as stated in Biot's studies (Biot, 1956a).        
Generally, the stresses acting on a porous medium can be separated into two parts: one is on the solid frame which can be written as σijs; the other is on the fluid represented by sij=−φpδij. Thus the total stresses are expressed by: σij=σijs+sij. Where φ is the porosity of the medium; p is the fluid pressure; δij is Kronecker symbol; the negative sign existing in the equation is for the association of directions between fluid pressure and stress. Starting with the above stress expressions of a porous medium and by employing the force equilibrium relation, the dynamics equations of a porous medium can be written as:
                    {                                                                                                  N                    ⁢                                                                  ∇                        2                                            ⁢                      u                                                        +                                      ∇                                          [                                                                                                    (                                                          A                              +                              N                                                        )                                                    ⁢                          e                                                +                                                  Q                          ⁢                                                                                                          ⁢                          ɛ                                                                    ]                                                                      =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            11                                                    ⁢                          u                                                +                                                                              ρ                            12                                                    ⁢                          U                                                                    )                                                        +                                      b                    ⁢                                          ∂                                              ∂                        t                                                              ⁢                                          (                                              u                        -                        U                                            )                                                                                                                                                                ∇                                      [                                          Qe                      +                                              R                        ⁢                                                                                                  ⁢                        ɛ                                                              ]                                                  =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            12                                                    ⁢                          u                                                +                                                                              ρ                            22                                                    ⁢                          U                                                                    )                                                        -                                      b                    ⁢                                          ∂                                              ∂                        t                                                              ⁢                                          (                                              u                        -                        U                                            )                                                                                                                              (                              1            ⁢            a                    ,          b                )            
The coefficient b is related to Darcy's coefficient of permeability k by
                    b        =                              μϕ            2                    k                                    (        2        )            where, μ is the fluid viscosity and φ is the porosity of the medium.
In Eq. (1), u and U are the displacement vectors of fluid and solid respectively, which consist of the quantities and directions of the displacements. While e and ε are the volume strains of the solid and fluid respectively with the expressions: e=∇·u, ε=∇·U·ρ11, ρ12 and ρ22 are density terms, which can be expressed as: ρ11=(1−φ)ρs, ρ22=φρf, ρ12=−(α−1)φρf, while ρs is the mass density of the solid grains, ρf is the mass density of the fluid in pores, α=(½)[φ−1+1], φ is the porosity of the medium. A, N, Q and R are the physical parameters of the medium. A and N are similar as Lame coefficients in elastic theory. N represents the shear modulus of the medium; R is a measure of pressure on the fluid required to drive a unit volume of fluid into the porous medium. Q describes the coupling between the volume change of solid and that of fluid. The expressions for A, N, Q and R will be given in following section.
Based on Eq. (1), Biot (1956a; 1956b) presented the expressions for three waves existing in a porous medium in the form of the volume strain. However, it is not convenient to quantify the displacements from volume strains, especially when a two- or three-dimensional domain is considered. Accordingly, the detailed description for deriving the waves expressions in the form of displacement will be present.
Applying Helmholtz decomposition to the displacement vectors of solid and fluid, respectively:
                    {                                                            u                =                                                      grad                    ⁡                                          (                                              φ                        s                                            )                                                        +                                      curl                    ⁡                                          (                                              ψ                        s                                            )                                                                                                                                              U                =                                                      grad                    ⁡                                          (                                              φ                        f                                            )                                                        +                                      curl                    ⁡                                          (                                              ψ                        f                                            )                                                                                                                              (                              3            ⁢            a                    ,          b                )            where φs and φf are scalar potentials of solid and fluid respectively, ψs and ψf are vector potentials for the displacements of solid and fluid. ψs and ψf also satisfy the conditions: ∇·ψs=0 and ∇·ψf=0.
For P-wave, also named compressional wave, the displacement is corresponding to the scalar potentials, without rotation, that implies ∇×u=0. For S-wave, also known as rotational wave or shear wave, the displacement is due to vector potentials, ∇·u=0. Substituting Eq. (3) into Eq. (1), and rearranging the terms according to the scalar and vector potentials, as Lin et al. (2001, Report No. CE 01-04, Los Angeles, Calif., USA) did in their research, two sets of equations can be obtained corresponding to scalar potentials and vector potentials of the fluid and solid. Thus, the expressions for P- and S-waves can be given as:
For P-wave:
                    {                                                                                                  ∇                    2                                    ⁢                                      (                                                                  P                        ⁢                                                                                                  ⁢                                                  φ                          s                                                                    +                                              Q                        ⁢                                                                                                  ⁢                                                  φ                          f                                                                                      )                                                  =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            11                                                    ⁢                                                      φ                            s                                                                          +                                                                              ρ                            12                                                    ⁢                                                      φ                            f                                                                                              )                                                        +                                      b                    ⁢                                          ∂                                              ∂                        t                                                              ⁢                                          (                                                                        φ                          s                                                -                                                  φ                          f                                                                    )                                                                                                                                                                                    ∇                    2                                    ⁢                                      [                                                                  Q                        ⁢                                                                                                  ⁢                                                  φ                          s                                                                    +                                              R                        ⁢                                                                                                  ⁢                                                  φ                          f                                                                                      ]                                                  =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            12                                                    ⁢                                                      φ                            s                                                                          +                                                                              ρ                            22                                                    ⁢                                                      φ                            f                                                                                              )                                                        -                                      b                    ⁢                                          ∂                                              ∂                        t                                                              ⁢                                          (                                                                        φ                          s                                                -                                                  φ                          f                                                                    )                                                                                                                              (                              4            ⁢            a                    ,          b                )            For S-wave:
                    {                                                                              N                  ⁢                                                            ∇                      2                                        ⁢                                          ψ                      s                                                                      =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            11                                                    ⁢                                                      ψ                            s                                                                          +                                                                              ρ                            12                                                    ⁢                                                      ψ                            f                                                                                              )                                                        +                                      b                    ⁢                                          ∂                                              ∂                        t                                                              ⁢                                          (                                                                        ψ                          s                                                -                                                  ψ                          f                                                                    )                                                                                                                                              0                =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            12                                                    ⁢                                                      ψ                            s                                                                          +                                                                              ρ                            22                                                    ⁢                                                      ψ                            f                                                                                              )                                                        -                                      b                    ⁢                                          ∂                                              ∂                        t                                                              ⁢                                          (                                                                        ψ                          s                                                -                                                  ψ                          f                                                                    )                                                                                                                              (                              5            ⁢            a                    ,          b                )            in which, P=A+2N is an introduced variable. Eqs. (4) and (5) are the governing equations of the waves propagating in porous media in terms of displacement potentials. These make it available to study the compression waves and shear wave separately or jointly in analyzing waves propagating in porous medium.
As in the case of purely elastic waves, the body waves can be separated into uncoupled rotational and dilatational waves. For P-wave, to get the governing equations expressed in the form of displacements, applying the divergence operation to Eq. (4), the equations for dilatational waves can be obtained in the following form:
                    {                                                                              ∇                                      [                                                                  ∇                        2                                            ⁢                                              (                                                                              P                            ⁢                                                                                                                  ⁢                                                          φ                              s                                                                                +                                                      Q                            ⁢                                                                                                                  ⁢                                                          φ                              f                                                                                                      )                                                              ]                                                  =                                                                                                          ∇                                      [                                                                                            ∂                          2                                                                          ∂                                                      t                            2                                                                                              ⁢                                              (                                                                                                            ρ                              11                                                        ⁢                                                          φ                              s                                                                                +                                                                                    ρ                              12                                                        ⁢                                                          φ                              f                                                                                                      )                                                              ]                                                  +                                  ∇                                      [                                          b                      ⁢                                              ∂                                                  ∂                          t                                                                    ⁢                                              (                                                                              φ                            s                                                    -                                                      φ                            f                                                                          )                                                              ]                                                                                                                                            ∇                                      [                                                                  ∇                        2                                            ⁢                                              (                                                                              Q                            ⁢                                                                                                                  ⁢                                                          φ                              s                                                                                +                                                      R                            ⁢                                                                                                                  ⁢                                                          φ                              f                                                                                                      )                                                              ]                                                  =                                                                                                          ∇                                      [                                                                                            ∂                          2                                                                          ∂                                                      t                            2                                                                                              ⁢                                              (                                                                                                            ρ                              12                                                        ⁢                                                          φ                              s                                                                                +                                                                                    ρ                              22                                                        ⁢                                                          φ                              f                                                                                                      )                                                              ]                                                  -                                  ∇                                      [                                          b                      ⁢                                              ∂                                                  ∂                          t                                                                    ⁢                                              (                                                                              φ                            s                                                    -                                                      φ                            f                                                                          )                                                              ]                                                                                                          (                              6            ⁢            a                    ,          b                )            
Let φ be a general displacement scalar potential and u a general displacement vector. For P-wave, the displacement vector u is just related to the scalar potential φ by:u=∇φ  (7)The scalar potential φ also has the following property:∇(∇2φ)=∇[∇·(∇φ)]=∇×[∇×(∇φ)]+∇2(∇φ)=∇2(∇φ)  (8)
Therefore, with equations of Eqs. (7) and (8), the governing equations of Eq. (4) for the dilatation waves can be written in the form of displacements as:
                    {                                                                                                  ∇                    2                                    ⁢                                      (                                                                  Pu                        sp                                            +                                              QU                        fp                                                              )                                                  =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            11                                                    ⁢                                                      u                            sp                                                                          +                                                                              ρ                            12                                                    ⁢                                                      U                            fp                                                                                              )                                                        +                                                                                                        b                ⁢                                  ∂                                      ∂                    t                                                  ⁢                                  (                                                            u                      sp                                        -                                          U                      fp                                                        )                                                                                                                                              ∇                    2                                    ⁢                                      [                                                                  Qu                        sp                                            +                                              RU                        fp                                                              ]                                                  =                                                                                                    ∂                        2                                                                    ∂                                                  t                          2                                                                                      ⁢                                          (                                                                                                    ρ                            12                                                    ⁢                                                      u                            sp                                                                          +                                                                              ρ                            22                                                    ⁢                                                      U                            fp                                                                                              )                                                        -                                                                                                        b                ⁢                                  ∂                                      ∂                    t                                                  ⁢                                  (                                                            u                      sp                                        -                                          U                      fp                                                        )                                                                                        (                              9            ⁢            a                    ,          b                )            in which, the subscript ‘s’ represents the displacement of solid, ‘f’ represents the displacement of the fluid, ‘p’ represents the displacement due to the P-wave. In Eq. (9), the parameters of material, P, Q, R can be expressed as (Plona et al., 1984, IN Physics and Chemistry of Porous Media, Johnson and Sen, Eds. American Institute of Physics, New York, pp. 89-104; Biot et al., 1957, J. Appl. Mech. 24: 594-601; Lin et al., 2001, Report No. CE 01-04, Los Angeles, Calif., USA):
                    P        =                                                                                                  (                                          1                      -                      ϕ                                        )                                    ⁡                                      [                                          1                      -                      ϕ                      -                                                                        K                          b                                                                          K                          s                                                                                      ]                                                  ⁢                                  K                  s                                            +                              ϕ                ⁢                                                      K                    s                                                        K                    f                                                  ⁢                                  K                  b                                                                    1              -              ϕ              -                                                K                  b                                                  K                  s                                            +                              ϕ                ⁢                                                      K                    s                                                        K                    f                                                                                +                                    4              3                        ⁢            N                                              (        10        )                                Q        =                                            [                              1                -                ϕ                -                                                      K                    b                                                        K                    s                                                              ]                        ⁢            ϕ            ⁢                                                  ⁢                          K              s                                            1            -            ϕ            -                                          K                b                                            K                s                                      +                          ϕ              ⁢                                                K                  s                                                  K                  f                                                                                        (        11        )                                R        =                                            ϕ              2                        ⁢                          K              s                                            1            -            ϕ            -                                          K                b                                            K                s                                      +                          ϕ              ⁢                                                K                  s                                                  K                  f                                                                                        (        12        )            in which, φ is the porosity of the porous medium; Kf, Ks, Kb, N are property parameters of the material. Kf is the bulk modulus of the fluid; Ks is the bulk modulus of the solid; Kb is bulk modulus of the skeletal frame; N is the shear modulus of the skeletal frame. Eq. (9) are the governing equations for P-wave propagating in the porous medium. It should be noted that the wave equations are all written in terms of displacements of solid and fluid. The governing equations in terms of displacement for S wave also can be obtained by applying the curl operator to Eq. (5).